This page is partly a test of using WordPress LaTeX rendering to generate some math equations, in this case rotation matrices. For now, that’s all it is…
Start with a 2D identity matrix:
The trick is look at the matrix as two vertical (column) vectors, called i-hat and j-hat. These start off as unit vectors along the x-axis and y-axis, respectively, and the matrix says where they end up.
A matrix describes a transformation of (in this case) the 2D plane. Such a transformation might be a scaling, a rotation, a shear, or some combination. In all cases, the origin stays the same.
In the case of the identity matrix, i-hat ends up at (1,0) and j-hat at (0,1), which is exactly where they started. That’s what makes the identity matrix the identity matrix. Nothing changes!
A 2D rotation matrix looks like this:
The name, identifies it as a Rotation matrix. The superscript indicates that it’s a rotation in two dimensions, and the subscript that the rotation is about the z-axis. Also, R is a function that takes an angle, θ (theta).
The result is a matrix with four values based on the sine and cosine of θ. Given the following table, it’s easy to plug in values for four cases of angle θ:
So the first rotation matrix, a rotation of zero degrees, looks like this:
Which is the identity matrix. A rotation of zero degrees does nothing.
The other three matrices look like this:
Of course, other angles result in real values.
For instance, a 10-degree rotation matrix looks like this:
The values are just cos(10) and sin(10) plugged into their appropriate spots.
The 3D identity matrix:
That’s also the matrix for a rotation of zero about any axis.
The matrix for 3D rotation around the x-axis:
The three matrices for x-axis rotations of 90°, 180°, and 270°, look like:
The matrix for 3D rotation around the y-axis:
The matrix for 3D rotation around the z-axis:
The instances for specific rotations look much like the ones shown above.
The 4D identity matrix:
The matrix for 4D rotation around the Y and Z axes:
The dual-axis rotation allows rotation of all orientations of cubes in 4D space.
In a tesseract, the four cubes with extent in X or W shift between extending in those axes, while the other four cubes (who have both X and W extent), rotate around either their Y or Z axis (depending on which they have).
In a (3D) cube, 4D rotation rotates the X extent into the W dimension and back (reversed), which allows “flipping” a cube. Consider what the above matrix in 3D:
This truncated version shifts the X extent into the W dimension, but that dimension is cut off and not shown. The effect is that the X dimension reduces to zero (at 90°), restores in the negative (reversed) direction (at 180°), reduces to zero (at 270°), and finally restores positively (at 360°).
The matrix for 4D rotation around the X and Z axes:
Which is the tesseract rotation that seems to move the cubes along the Y-axis (the second mode of rotation seen in the video).
The matrix for 4D rotation around the X and Y axes:
Which is the tesseract rotation that seems to move cubes along the Z-axis (the third mode of rotation seen in the video).